For many years, I have been fascinated by complex 2-manifolds bordered by intricate smooth curves, as can be found in sculptures by Eva Hild or Charles Perry. Even on very simple arrangements of border curves, rather intriguing minimal surfaces can be supported. On the highly symmetrical Borromean arrangement of three oval borders, in which no pair of ovals are actually linked, Ken Brakke has identified 15 different possible soap films. The simplest one is a stable, a non-orientable minimal surface with the symmetry of the oriented tetrahedron. I am investigating sculptures that result when multiple copies of this structure are recursively nested inside one another.
I start with the simplest Borromean soap film surface and place a
smaller copy inside so that they just touch. To obtain a proper
border configuration for a 2-manifold, I transform the six areas
where two ovals touch into skewed crossing of two smooth curves.
This results in a border curve system consisting of six simple
interlinked loops. I force these loops to be perfectly circular,
and then construct a soap-film on this border structure.
To enhance the transparency of this sculpture and allow a better look at the inside geometry, I also cut out eight small circular holes from the two sets of 3-sided face patches in the outer and inner levels of the soap film. The 2-manifold is single-sided, of genus 6, and has 6+8 punctures.
Here I am placing three of the simple Borromean soap film surfaces inside one another. Again, I want to turn this into a single 2-manifold surface with a cohesive, smooth border curve structure. In this case, when the 12 touch-points of the 9 scaled ovals are turned into skewed crossings of smooth curves, the three ovals that were lying in the same plane readily turn into a single simple knot. This is either a Figure-8 knot or a (2,3)-Torus knot, depending on the choice of subsequent over- and under-passes. The torus knot will lead to smoother border curves, since it requires less undulations in the 3rd dimension than the alternating Figure-8 knot. The resulting single-sided 2-manifold has 3+12 borders and is of genus 9.